
\softpara{\textbf{Model \& Notations}}

The spectrum division problem in fact can be modeled as a more general power signal assigning problem. Consider a set of wireless link with maximum
power limit and a given spectrum. The part of the spectrum used by each link and the amount of power is in fact a power signal in the given spectrum
where the signal is zero in part of the bandwidth which is not used by the link. Hence, we focus on the more general power signal assignment problem
as defined subsequently.

In this paper, we assume that we are given a set of wireless links and we have an available spectrum. We assume
that each sender has the flexibility to produce any shape of power signal along the spectrum as long as the total
used power is below the nodes limit. The throughput of each link is computed by generalized Shannon-Hartly theorem
for frequency dependent noise. These assumption would let us compute the theoretical limit of a system transmission
capacity with the given limits on power and available frequency.  The solution obtained using the theoretical limit
is rounded up to fit the current available hardware for experiments. We use the following notations throughout the paper.

Assume we are given a set of $n$ wireless links each with a sender and a receiver in a 2D-plane and spectrum $B$ with the following parameters.
\begin{itemize}
    \item \textbf{$p_i$} is the maximum power of sender of link $i$.
    \item \textbf{$N_i$} is the background noise per hertz levels at receiver $i$.
     We assume that the background noise is constant across the given spectrum for each receiver but might vary among receivers. 
%    \item \textbf{$p_{ij}$} is the signal strength of sender $i$ at receiver $j$ and a spectrum $B$.
    \item \textbf{$d_{ij}$} is the distance of sender $i$ and receiver $j$.
    \item \textbf{$C_i$} is the capacity of each link as computed by Shannon-Hartley theorem. $C_i$ is computed using the following formula.
        \[
            C_i= \int\limits_{{{b}_{1}}}^{{{b}_{2}}}{\log \left( 1+\frac{S(f)}{N(f)} \right)df}
        \]
        where $(b_1,b_2)$ is the available spectrum, $S(f)$ is the signal strength of sender of link $i$ at the receiver of the link and $N(f)$ 
        is the noise plus interference at receiver of link $i$.
    \item \textbf{$\alpha$} is the path loss exponent.
\end{itemize}


\softpara{\textbf{Problem Definitions}}
In this paper, we focus on problem of assigning a power signal to each of the links maximizing capacity while maintaining fairness. This is a dual objective problem and there has been extensive discussion on the community on how to define a single measure encapsulating both objectives. We use three widely accepted measures, Maximum Total Throughput, Maximum Proportional Fairness and maximum minimum throughput as our measures. We formally define our problem as below.

\textbf{Power Assignment Problem:}
We are given a system $n$ wireless links, with $p_i$ is the power limit of link $i$ and $d_{ij}$ is the distance of sender of link $i$ and reciever of link $j$. For a given assignment of power signals to each link, assume $C_i$ is the throughput of each link computed by Shannon-Hartley theorem for a given path loss ratio $\alpha$. Maximize one of the following measures.
\begin{itemize}
\item Max Total Throughput (MTT): \[Max\left(\sum_{i=1}^{n}C_{i}\right)\].

\item Max Proportional Fairness  (MPF):\[Max\left(\Pi_{i=1}^{n}C_{i}\right)\].

\item Max-Min Throughput (MMT): \[Max\left(min_{i=1..n}(C_{i})\right)\].
\end{itemize}

Throughout the rest of this paper, we refer to the problem with the three objectives as MTT, MPF and MMT respectively.

The rest of the paper is organized as follows. We first study the power assignment problem in the special case of two wireless links.
We present the major result of the paper. \emph{We prove that optimum solution would not use partial overlapping channels}. We then solve the problem for
two wireless links. We then, consider the case of $n$ wireless links. We prove that the problem is NP-hard and then we provide a
heuristic algorithm based on the result that we got from special case of two wireless links. We then consider a special case of
the problem, where the length of each link is limited and nodes location has uniform distribution. We compute an upper bound
for the optimum solution. We use the computed upperbound to measure the performance of our heuristic algorithm. We then run real
experiments to compare the throughput achieved by using our algorithm to the throughput achieved through conventional channel assignment techniques.